From: Snapshot-Content-Location: file:///C:/Users/wille/OneDrive/Desktop/WGU/R%20Course%20Materials/wineproject.html Subject: wineproject.knit Date: Tue, 10 Aug 2021 03:04:48 -0000 MIME-Version: 1.0 Content-Type: multipart/related; type="text/html"; boundary="----MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC----" ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/html Content-ID: Content-Transfer-Encoding: quoted-printable Content-Location: file:///C:/Users/wille/OneDrive/Desktop/WGU/R%20Course%20Materials/wineproject.html wineproject.knit

Exploration Of Red Wine Quality
by
Will Everhard


This data is based off of a study done on wine quality. The original stu= dy was on both red and white wines but I will be using the red wine part fo= r the sake of keeping the study smaller. Directly from the documentation: = =E2=80=9CThe inputs include objective tests (e.g. PH values) and the o= utput is based on sensory data (median of at least 3 evaluations made by wi= ne experts). Each expert graded the wine quality between 0 (very bad) and 1= 0 (very excellent).=E2=80=9D

=E2=80=9CThe dataset is related to red variants of the Portuguese=E2=80= =9DVinho Verde" wine. For more details, consult: http://www.vinhoverde.pt/en/ or the refere= nce [Cortez et al., 2009]. Due to privacy and logistic issues, only physico= chemical (inputs) and sensory (the output) variables are available (e.g.&nb= sp;there is no data about grape types, wine brand, wine selling price, etc.= )."

Description of the variables:

##  [1] "X"                    "fixed.acidity"   =
     "volatile.acidity"   =20
##  [4] "citric.acid"          "residual.sugar"       "chlorides"          =
=20
##  [7] "free.sulfur.dioxide"  "total.sulfur.dioxide" "density"            =
=20
## [10] "pH"                   "sulphates"            "alcohol"            =
=20
## [13] "quality"
  • Fixed acidity: most acids involved with wine or fixed or nonvolatile= (do not evaporate readily).
  • Volatile acidity: the amount of acetic acid in wine, which at too hi= gh of levels can lead to an unpleasant, vinegar taste.
  • Citric acid: found in small quantities, citric acid can add =E2=80= =98freshness=E2=80=99 and flavor to wines.
  • Residual sugar: the amount of sugar remaining after fermentation sto= ps, it=E2=80=99s rare to find wines with less than 1 gram/liter and wines w= ith greater than 45 grams/liter are considered sweet.
  • Chlorides: the amount of salt in the wine.
  • Free sulfur dioxide: the free form of SO2 exists in equilibrium betw= een molecular SO2 (as a dissolved gas) and bisulfite ion; it prevents micro= bial growth and the oxidation of wine.
  • Total sulfur dioxide: amount of free and bound forms of S02; in low = concentrations, SO2 is mostly undetectable in wine, but at free SO2 concent= rations over 50 ppm, SO2 becomes evident in the nose and taste of wine.
  • Density: the density of water is close to that of water depending on= the percent alcohol and sugar content.
  • pH: describes how acidic or basic a wine is on a scale from 0 (very = acidic) to 14 (very basic); most wines are between 3-4 on the pH scale.
  • Sulphates: a wine additive which can contribute to sulfur dioxide ga= s (S02) levels, wich acts as an antimicrobial and antioxidant.
  • Alcohol: the percent alcohol content of the wine.
  • Quality: (score between 0 and 10) - an output variable based on sens= ory data.

After looking at the variables, I noticed there is an extra variable nam= ed =E2=80=9CX=E2=80=9D. I didn=E2=80=99t see any documentation on this so I= will keep that in my mind as I investigate.

Here, I further investigate the variable types=E2=80=A6

## 'data.frame':    1599 obs. of  13 variables:
##  $ X                   : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ fixed.acidity       : num  7.4 7.8 7.8 11.2 7.4 7.4 7.9 7.3 7.8 7.5 .=
..
##  $ volatile.acidity    : num  0.7 0.88 0.76 0.28 0.7 0.66 0.6 0.65 0.58 =
0.5 ...
##  $ citric.acid         : num  0 0 0.04 0.56 0 0 0.06 0 0.02 0.36 ...
##  $ residual.sugar      : num  1.9 2.6 2.3 1.9 1.9 1.8 1.6 1.2 2 6.1 ...
##  $ chlorides           : num  0.076 0.098 0.092 0.075 0.076 0.075 0.069 =
0.065 0.073 0.071 ...
##  $ free.sulfur.dioxide : num  11 25 15 17 11 13 15 15 9 17 ...
##  $ total.sulfur.dioxide: num  34 67 54 60 34 40 59 21 18 102 ...
##  $ density             : num  0.998 0.997 0.997 0.998 0.998 ...
##  $ pH                  : num  3.51 3.2 3.26 3.16 3.51 3.51 3.3 3.39 3.36=
 3.35 ...
##  $ sulphates           : num  0.56 0.68 0.65 0.58 0.56 0.56 0.46 0.47 0.=
57 0.8 ...
##  $ alcohol             : num  9.4 9.8 9.8 9.8 9.4 9.4 9.4 10 9.5 10.5 ..=
.
##  $ quality             : int  5 5 5 6 5 5 5 7 7 5 ...

There are 1599 observations (wines) and 13 variables in this study with = all of them set to num except for the variables =E2=80=9CX=E2=80=9D and =E2= =80=9Cquality=E2=80=9D.

Since the point of the data article is focused on quality, I will single= out quality first=E2=80=A6

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max=
.=20
##   3.000   5.000   6.000   5.636   6.000   8.000

It appears that quality ranges from 3 to 8 even though the study says it= goes from 1 to 10.

Now I will check out a summary of the rest of the data=E2=80=A6

##        X          fixed.acidity   volatile.aci=
dity  citric.acid  =20
##  Min.   :   1.0   Min.   : 4.60   Min.   :0.1200   Min.   :0.000 =20
##  1st Qu.: 400.5   1st Qu.: 7.10   1st Qu.:0.3900   1st Qu.:0.090 =20
##  Median : 800.0   Median : 7.90   Median :0.5200   Median :0.260 =20
##  Mean   : 800.0   Mean   : 8.32   Mean   :0.5278   Mean   :0.271 =20
##  3rd Qu.:1199.5   3rd Qu.: 9.20   3rd Qu.:0.6400   3rd Qu.:0.420 =20
##  Max.   :1599.0   Max.   :15.90   Max.   :1.5800   Max.   :1.000 =20
##  residual.sugar     chlorides       free.sulfur.dioxide total.sulfur.dio=
xide
##  Min.   : 0.900   Min.   :0.01200   Min.   : 1.00       Min.   :  6.00  =
   =20
##  1st Qu.: 1.900   1st Qu.:0.07000   1st Qu.: 7.00       1st Qu.: 22.00  =
   =20
##  Median : 2.200   Median :0.07900   Median :14.00       Median : 38.00  =
   =20
##  Mean   : 2.539   Mean   :0.08747   Mean   :15.87       Mean   : 46.47  =
   =20
##  3rd Qu.: 2.600   3rd Qu.:0.09000   3rd Qu.:21.00       3rd Qu.: 62.00  =
   =20
##  Max.   :15.500   Max.   :0.61100   Max.   :72.00       Max.   :289.00  =
   =20
##     density             pH          sulphates         alcohol    =20
##  Min.   :0.9901   Min.   :2.740   Min.   :0.3300   Min.   : 8.40 =20
##  1st Qu.:0.9956   1st Qu.:3.210   1st Qu.:0.5500   1st Qu.: 9.50 =20
##  Median :0.9968   Median :3.310   Median :0.6200   Median :10.20 =20
##  Mean   :0.9967   Mean   :3.311   Mean   :0.6581   Mean   :10.42 =20
##  3rd Qu.:0.9978   3rd Qu.:3.400   3rd Qu.:0.7300   3rd Qu.:11.10 =20
##  Max.   :1.0037   Max.   :4.010   Max.   :2.0000   Max.   :14.90 =20
##     quality    =20
##  Min.   :3.000 =20
##  1st Qu.:5.000 =20
##  Median :6.000 =20
##  Mean   :5.636 =20
##  3rd Qu.:6.000 =20
##  Max.   :8.000

I am interested in how variables correlate with quality.

Acids related thoughts: Acidity level is determined by pH = levels. The lower the level of pH, the higher the level of acidity. For exa= mple, stomach acid is 1.0 and antacids have a pH of 10.5 The average pH lev= els for red wines are between 3.5 and 3.8.

Just like the density, the statistics observed here for pH levels relati= vely match the norm with the minimum at 2.740, the mean at 3.311, the media= n at 3.310, and the max at 4.010.

Density related thoughts: Density in wine is measured with= a hydrometer. Water has the density of 1 and that is what is used to contr= ast the density of wine. The density of wine juice is higher than water bec= ause there are sugars and other things like pigmentations in it. The typica= l density of the must (the term used for the juice before the yeast is adde= d) is generally between 1.080 and 1.090. This essentially means wine is 8-9= % more dense than water.

Alcohol is approximately 0.8, or 20% less dense than water. As the yeast= consumes the sugar in the wine, and converts it to alcohol, the must gradu= ally becomes less dense. After fermentation is complete, the density of win= e should be roughly at, or slightly less than 1.00, often 0.996.

Comparatively speaking, the statistics observed here relatively match th= e norm with the minimum at 0.9901, the mean at 0.9967, the median at 0.9968= , and the max at 1.0037.

None of the wines are near the article=E2=80=99s stated level of sweet w= ines at a residual sugar level of 45 grams per liter so sweet wines can be = ruled out of this exploration.

Sulfurs related thoughts: It is generally widely accepted = amongst wine drinkers that an aged wine is associated with =E2=80=9Cbetter= =E2=80=9D wine. Sulfites in wine is naturally occurring and is also added t= o prevent microbial growth and oxidation of the wine but with too much sulf= ites, it can have an onion or egg yolk hint of smell to it. Sulfites dissip= ate over time, this may be a correlation to the age and quality of wine.

It has also been said in an article about sulfites in wine, that, =E2=80= =9CNatural Wines have an authenticity of taste that most modern wines have = lost. They=E2=80=99re complex, unusual, surprising, joyful. They leap out o= f the glass with a vivacity that=E2=80=99s far too rare in today=E2=80=99s = winemaking world. When you drink Natural Wine, you can taste its origin, it= s terroir, and all the subtleties that make it unique. That=E2=80=99s only = possible in low-sulfite wines.=E2=80=9D

I was not really able to find a quick enough explanation of how this dat= a measured sulphates so I use this information to see how sulphates relate = to higher quality wines.

Salty thoughts: As far as salinity goes, salt can accentua= te aromatics from a terrior by the taster associating the smells with the t= aste of salt and sometimes a saltier wine is looked for in those situations= . However, Roman Roth, winemaker at W=C3=B6lffer Estate in Sagaponack, New = York, says that a good wine should always have =E2=80=9Ca number of things = competing for your attention. Is it acidity? Is it minerality? Is it tannin= s, or creamy yeast characters? Is it salinity? Nothing should stand out. Th= ey should all be in a harmonious balance, making the wine interesting and g= iving it finesse.=E2=80=9D

I was not able to find a reliable source in a timely manner for a standa= rd of when a wine becomes =E2=80=9Ctoo=E2=80=9D salty. I will look for a ge= neral balance of chloride in relation to quality considering the article I = quoted.

Univariate Plots Section

I=E2=80=99ll start with quality since it is the title of the data articl= e.

There appears to be a normal distribution with a very small rating range= . Considering most of the wines ranked within the qualities of 5 & 6, I= will create 3 ranges from the data ratings as Low Quality, Average, and Hi= gh Quality to observe the data more closely.

Univariate Analysis

There is a great deal of average quality red wines (the quality ratings = of 5 and 6). There are a lot of low alcohol wines around 9%. The sulphates = range seems pretty low going from 0.5 to 2.0 with most of the sulphate leve= ls just above 0.5. pH and Density have a normal distribution with their ave= rages being in the middle of the ranges. Total sulfur dioxide, free sulfur = dioxide, chlorides, residual sugars, and fixed acids all seem to have the m= ajorities on the lower end of their ranges. Citric acid and volatile acid h= ave the most unique distributions showing almost bimodal and right skewed.<= /p>

I am mostly interested in looking at the acids since they have the most = variance. I will contrast with other variables against quality as well.

I turned quality into a factor and combined levels to get a larger pool = of samples for high quality and low quality wines since the majority of the= samples are within 5 and 6.

Bivariate Plots Section

High alcohol had the highest positive correlation I came across but it w= as weird in that it was lowest at the middle before spiking high in higher = quality wines. I had to take a closer look by using the original quality se= paration of red wines. Funny enough the lowest mean is at 5 while it also h= olds the highest levels of alcohol recorded in the observations as outliers= .

Sulphates have a steady positive correlation with quality.

pH has a slight negative correlation with quality.

Density seems to have little to no effect on quality though it is slight= ly lower in quality wines.

Total sulfur dioxide has no visible effect on quality.

Free sulfur dioxides have no effect on quality.

Chlorides have at best, minimal to no effect on quality.

Residual sugar seems to have no effect on quality.

Citric acid shows a clear positive correlation to quality.

Volatile acidity shows a clear negative correlation with quality.

Fixed acidity seems to have a positive correlation with quality.

This plot confuses me a bit. High pH means low acidity yet this plot sug= gests a positive correlation. Volatile does mean unstable though so I can m= ake sense of the acidity dissipating.

This contrast makes perfect sense with pH and fixed acidity inversely co= rrelated.

This plot agrees with acidity inversely correlating with pH levels.

Citric acid and sulphates show a positive correlation.

pH and sulphates show a slight negative correlation.

This plot makes sense of the pH vs sulphates as acidity is the opposite = of pH there is and should be a slight positive correlation here.

This plot further agrees with earlier findings of volatile acidity actin= g more basic in measure than acidic.

Bivariate Analysis

Volatile acidity looks to be the most inversely correlated with quality.= .

Citric acid, fixed acidity, sulphates, and alcohol have the best positiv= e correlations with quality.

High levels of alcohol seems to have the strongest correlation with high= quality red wines.

Multivariate Plots Section

This graph surprises me in that it suggests a positive correlation betwe= en alcohol and pH. Though in the higher quality wines we see that pH caps o= ff at 3.75 while low quality and average go higher in pH.

Overall we see a negative correlation between alcohol and fixed acidity = but when looking at the different levels of quality, higher quality wines h= ave higher alcohol and higher fixed acidity when compared to lower levels o= f quality. I do not see a strong relationship here and wine is generally ac= idic.

Here we see that alcohol goes higher with quality and volatile acidity g= oes lower with quality. They have an inverse relationship towards each othe= r when determining quality.

Here we see a positive correlation with citric acid and alcohol in regar= ds to quality. I notice in high quality wines that citric acid tops off aro= und 0.75. It does show some of the high quality wines having almost no citr= ic acid at all. I speculate that citric acid is more of a popular personal = taste of certain varietals than a 100% clear decisive determining factor of= quality wine.

This graph shows a slight positive correlation between alcohol and sulph= ates. I=E2=80=99m noticing here that sulphates in the higher quality cap of= f lower than the other levels of quality.

I see a positive correlation between citric acid and sulphates but I als= o see that the range of sulphates shrinks as the quality rises.

I see a negative correlation between alcohol and density but the combina= tion of alcohol and density rises with quality.

This graph clearly shows high alcohol and low residual sugars with high = quality red wines.

Density has a varying range amongst higher quality wines while residual = sugar remains low.

The higher the sulphates, the lower the volatile acidity. This makes sen= se because sulphates help prevent oxidation. However, sulphates are higher = on average in high quality wines but the range of sulphates shrinks with hi= gher quality and is still on the lower side overall.

pH seems to have little to no effect on quality while sulphates remain t= he same in high quality wines with a small range averaging on lower levels.=

Sulphates contrasted with fixed acidity seems to have little to no effec= t on quality.

Multivariate Analysis

Alcohol seemed overall to strengthen everything with some help from citr= ic acid and sulphates. However, alcohol and citric acid capped off at a med= ium to high level where sulphates in the highest quality wines were low eve= n though sulphates helped to raise the quality.

I was surprised to see that density of wine in high quality wines had a = positive effect since residual sugars had a negative effect and high sugar = is usually associated with high density while alcohol makes wine less dense= . I wonder if concentration of grape skin pigments make for more dense more= aromatic wine while residual sugar stays low.

Final Plots and Summary

Plot One

Here we see alcohol greatly raises the quality of wines with the help of= citric acid. Citric acid caps off lower in the high quality wines compared= to the other levels of wine quality but the numbers cluster above mid leve= l for citric acid. That being said, citric acid doesn=E2=80=99t make great = wine by itself but it certainly seems to be a popular variable while it is = within a certain range when deciding if it is a high quality wine

Plot Two

Here we see sulphates playing a role similar to citric acid in that it i= sn=E2=80=99t maximized in high quality wines but just the right amount, wit= hin a small range, elevates the wine quality along with alcohol.

Plot Three

Here we see citric acid and sulphates positively effecting the quality o= f wine but within a small range compared to average and low quality.

Reflection

Overall, I found that alcohol plays a great deal in raising red wine qua= lity along with a little help from citric acid, density, and sulphates. Obv= iously, alcohol alone does not determine a high quality wine. Some things t= o consider from looking at these results are that alcohol is volatile and t= herefor can fume aromatics more potently. Low levels of sugar with high qua= lity wines yet density increasing with alcohol and quality suggests other e= lements, like, perhaps the concentration/thickness of the grape skins that = give it high tannins, deep color, and aromatics.

Another assumption I deduce from seeing low levels of sulphates in high = quality wines but the need for them to be present in high quality wines is,= we all know aged wines are generally considered =E2=80=9Chigher quality=E2= =80=9D wine, so I suggest the sulphates did their job well preserving the w= ine and as the wine aged the sulphates dissipated. Wines that never had eno= ugh sulphates in them probably oxidized early on and became less favored. W= ines that had a lot of sulphates probably had too much added or where not a= ged well enough for the sulphates to dissipate. It seems the quote at the b= eginning of my analysis rings true that high quality wines are well balance= d.

I think the most difficult part of this analysis was to focus in on cert= ain points and not over thinking it. Data can be shown in many ways and sti= ll be made sense of so it was difficult having such a variety to choose fro= m. For further analysis of red wine quality, new variables like grape varie= tal, terroir, harvest season, statistics on the weather of each year the gr= apes where harvested, etc. could make a significant impact on determining w= hat variables makes wine quality high.

------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-2d63dec9-c2ab-4733-b8ba-12af14695ac9@mhtml.blink @charset "utf-8"; h1 { font-size: 34px; } h1.title { font-size: 38px; } h2 { font-size: 30px; } h3 { font-size: 24px; } h4 { font-size: 18px; } h5 { font-size: 16px; } h6 { font-size: 12px; } code { color: inherit; background-color: rgba(0, 0, 0, 0.04); } pre:not([class]) { background-color: white; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-e1c5f508-6f94-4730-a8d9-e3bbe5d3189d@mhtml.blink @charset "utf-8"; code { white-space: pre-wrap; } span.smallcaps { font-variant: small-caps; } span.underline { text-decoration: underline; } div.column { display: inline-block; vertical-align: top; width: 50%; } div.hanging-indent { margin-left: 1.5em; text-indent: -1.5em; } ul.task-list { list-style: none; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-0bce418d-6364-41f5-a8ce-227c465d028b@mhtml.blink @charset "utf-8"; code { white-space: pre; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-0baa5026-47d2-4a26-b19d-eee669b323bd@mhtml.blink @charset "utf-8"; .main-container { max-width: 940px; margin-left: auto; margin-right: auto; = } img { max-width: 100%; } .tabbed-pane { padding-top: 12px; } .html-widget { margin-bottom: 20px; } button.code-folding-btn:focus { outline: none; } summary { display: list-item; } pre code { padding: 0px; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-60c41ec5-de91-41db-ab4b-506b87a2bf4d@mhtml.blink @charset "utf-8"; .tabset-dropdown > .nav-tabs { display: inline-table; max-height: 500px; mi= n-height: 44px; overflow-y: auto; border: 1px solid rgb(221, 221, 221); bor= der-radius: 4px; } .tabset-dropdown > .nav-tabs > li.active::before { content: "=EE=89=99"; fo= nt-family: "Glyphicons Halflings"; display: inline-block; padding: 10px; bo= rder-right: 1px solid rgb(221, 221, 221); } .tabset-dropdown > .nav-tabs.nav-tabs-open > li.active::before { content: "= =EE=89=98"; border: none; } .tabset-dropdown > .nav-tabs.nav-tabs-open::before { content: "=EE=89=99"; = font-family: "Glyphicons Halflings"; display: inline-block; padding: 10px; = border-right: 1px solid rgb(221, 221, 221); } .tabset-dropdown > .nav-tabs > li.active { display: block; } .tabset-dropdown > .nav-tabs > li > a, .tabset-dropdown > .nav-tabs > li > = a:focus, .tabset-dropdown > .nav-tabs > li > a:hover { border: none; displa= y: inline-block; border-radius: 4px; background-color: transparent; } .tabset-dropdown > .nav-tabs.nav-tabs-open > li { display: block; float: no= ne; } .tabset-dropdown > .nav-tabs > li { display: none; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-a0702a0a-a2ee-44d9-82e7-701d6ea0011d@mhtml.blink @charset "utf-8"; .MathJax_Hover_Frame { border-radius: 0.25em; box-shadow: rgb(136, 51, 170)= 0px 0px 15px; display: inline-block; position: absolute; border: 1px solid= rgb(170, 102, 221) !important; } .MathJax_Menu_Button .MathJax_Hover_Arrow { position: absolute; cursor: poi= nter; display: inline-block; border: 2px solid rgb(170, 170, 170); border-r= adius: 4px; font-family: "Courier New", Courier; font-size: 9px; color: rgb= (240, 240, 240); } .MathJax_Menu_Button .MathJax_Hover_Arrow span { display: block; background= -color: rgb(170, 170, 170); border: 1px solid; border-radius: 3px; line-hei= ght: 0; padding: 4px; } .MathJax_Hover_Arrow:hover { color: white !important; border: 2px solid rgb= (204, 204, 204) !important; } .MathJax_Hover_Arrow:hover span { background-color: rgb(204, 204, 204) !imp= ortant; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-a06ad71a-8b20-4227-a96e-1932cfa5c354@mhtml.blink @charset "utf-8"; #MathJax_About { position: fixed; left: 50%; width: auto; text-align: cente= r; border: 3px outset; padding: 1em 2em; background-color: rgb(221, 221, 22= 1); color: black; cursor: default; font-family: message-box; font-size: 120= %; font-style: normal; text-indent: 0px; text-transform: none; line-height:= normal; letter-spacing: normal; word-spacing: normal; overflow-wrap: norma= l; white-space: nowrap; float: none; z-index: 201; border-radius: 15px; box= -shadow: rgb(128, 128, 128) 0px 10px 20px; } #MathJax_About.MathJax_MousePost { outline: none; } .MathJax_Menu { position: absolute; background-color: white; color: black; = width: auto; padding: 2px; border: 1px solid rgb(204, 204, 204); margin: 0p= x; cursor: default; font: 400 12px "Segoe UI"; text-align: left; text-inden= t: 0px; text-transform: none; letter-spacing: normal; word-spacing: normal;= overflow-wrap: normal; white-space: nowrap; float: none; z-index: 201; box= -shadow: rgb(128, 128, 128) 0px 10px 20px; } .MathJax_MenuItem { padding: 2px 2em; background: transparent; } .MathJax_MenuArrow { position: absolute; right: 0.5em; padding-top: 0.25em;= color: rgb(102, 102, 102); font-size: 0.75em; } .MathJax_MenuActive .MathJax_MenuArrow { color: white; } .MathJax_MenuArrow.RTL { left: 0.5em; right: auto; } .MathJax_MenuCheck { position: absolute; left: 0.7em; } .MathJax_MenuCheck.RTL { right: 0.7em; left: auto; } .MathJax_MenuRadioCheck { position: absolute; left: 1em; } .MathJax_MenuRadioCheck.RTL { right: 1em; left: auto; } .MathJax_MenuLabel { padding: 2px 2em 4px 1.33em; font-style: italic; } .MathJax_MenuRule { border-top: 1px solid rgb(204, 204, 204); margin: 4px 1= px 0px; } .MathJax_MenuDisabled { color: graytext; } .MathJax_MenuActive { background-color: highlight; color: highlighttext; } .MathJax_MenuDisabled:focus, .MathJax_MenuLabel:focus { background-color: r= gb(232, 232, 232); } .MathJax_ContextMenu:focus { outline: none; } .MathJax_ContextMenu .MathJax_MenuItem:focus { outline: none; } #MathJax_AboutClose { top: 0.2em; right: 0.2em; } .MathJax_Menu .MathJax_MenuClose { top: -10px; left: -10px; } .MathJax_MenuClose { position: absolute; cursor: pointer; display: inline-b= lock; border: 2px solid rgb(170, 170, 170); border-radius: 18px; font-famil= y: "Courier New", Courier; font-size: 24px; color: rgb(240, 240, 240); } .MathJax_MenuClose span { display: block; background-color: rgb(170, 170, 1= 70); border: 1.5px solid; border-radius: 18px; line-height: 0; padding: 8px= 0px 6px; } .MathJax_MenuClose:hover { color: white !important; border: 2px solid rgb(2= 04, 204, 204) !important; } .MathJax_MenuClose:hover span { background-color: rgb(204, 204, 204) !impor= tant; } .MathJax_MenuClose:hover:focus { outline: none; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-3b3d2cf7-6610-4a79-b10d-b4f304e7ec2e@mhtml.blink @charset "utf-8"; .MathJax_Preview .MJXf-math { color: inherit !important; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-f3af06d2-d033-46ee-8d8c-a7fc39ea0f4c@mhtml.blink @charset "utf-8"; .MJX_Assistive_MathML { top: 0px; left: 0px; clip: rect(1px, 1px, 1px, 1px)= ; user-select: none; position: absolute !important; padding: 1px 0px 0px !i= mportant; border: 0px !important; height: 1px !important; width: 1px !impor= tant; overflow: hidden !important; display: block !important; } .MJX_Assistive_MathML.MJX_Assistive_MathML_Block { width: 100% !important; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-e7e7c11d-1f4b-46b9-9728-432340342dd0@mhtml.blink @charset "utf-8"; #MathJax_Zoom { position: absolute; background-color: rgb(240, 240, 240); o= verflow: auto; display: block; z-index: 301; padding: 0.5em; border: 1px so= lid black; margin: 0px; font-weight: normal; font-style: normal; text-align= : left; text-indent: 0px; text-transform: none; line-height: normal; letter= -spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space:= nowrap; float: none; box-sizing: content-box; box-shadow: rgb(170, 170, 17= 0) 5px 5px 15px; } #MathJax_ZoomOverlay { position: absolute; left: 0px; top: 0px; z-index: 30= 0; display: inline-block; width: 100%; height: 100%; border: 0px; padding: = 0px; margin: 0px; background-color: white; opacity: 0; } #MathJax_ZoomFrame { position: relative; display: inline-block; height: 0px= ; width: 0px; } #MathJax_ZoomEventTrap { position: absolute; left: 0px; top: 0px; z-index: = 302; display: inline-block; border: 0px; padding: 0px; margin: 0px; backgro= und-color: white; opacity: 0; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-36e8dd3b-3dd7-4040-829d-29ec9f38a0ae@mhtml.blink @charset "utf-8"; .MathJax_Preview { color: rgb(136, 136, 136); } #MathJax_Message { position: fixed; left: 1em; bottom: 1.5em; background-co= lor: rgb(230, 230, 230); border: 1px solid rgb(149, 149, 149); margin: 0px;= padding: 2px 8px; z-index: 102; color: black; font-size: 80%; width: auto;= white-space: nowrap; } #MathJax_MSIE_Frame { position: absolute; top: 0px; left: 0px; width: 0px; = z-index: 101; border: 0px; margin: 0px; padding: 0px; } .MathJax_Error { color: rgb(204, 0, 0); font-style: italic; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-81aba94b-9e10-480d-bfdc-d37870e345f3@mhtml.blink @charset "utf-8"; .MJXp-script { font-size: 0.8em; } .MJXp-right { transform-origin: right center; } .MJXp-bold { font-weight: bold; } .MJXp-italic { font-style: italic; } .MJXp-scr { font-family: MathJax_Script, "Times New Roman", Times, STIXGene= ral, serif; } .MJXp-frak { font-family: MathJax_Fraktur, "Times New Roman", Times, STIXGe= neral, serif; } .MJXp-sf { font-family: MathJax_SansSerif, "Times New Roman", Times, STIXGe= neral, serif; } .MJXp-cal { font-family: MathJax_Caligraphic, "Times New Roman", Times, STI= XGeneral, serif; } .MJXp-mono { font-family: MathJax_Typewriter, "Times New Roman", Times, STI= XGeneral, serif; } .MJXp-largeop { font-size: 150%; } .MJXp-largeop.MJXp-int { vertical-align: -0.2em; } .MJXp-math { display: inline-block; line-height: 1.2; text-indent: 0px; fon= t-family: "Times New Roman", Times, STIXGeneral, serif; white-space: nowrap= ; border-collapse: collapse; } .MJXp-display { display: block; text-align: center; margin: 1em 0px; } .MJXp-math span { display: inline-block; } .MJXp-box { text-align: center; display: block !important; } .MJXp-box::after { content: " "; } .MJXp-rule { margin-top: 0.1em; display: block !important; } .MJXp-char { display: block !important; } .MJXp-mo { margin: 0px 0.15em; } .MJXp-mfrac { margin: 0px 0.125em; vertical-align: 0.25em; } .MJXp-denom { width: 100%; display: inline-table !important; } .MJXp-denom > * { display: table-row !important; } .MJXp-surd { vertical-align: top; } .MJXp-surd > * { display: block !important; } .MJXp-script-box > * { height: 50%; display: table !important; } .MJXp-script-box > * > * { vertical-align: top; display: table-cell !import= ant; } .MJXp-script-box > :last-child > * { vertical-align: bottom; } .MJXp-script-box > * > * > * { display: block !important; } .MJXp-mphantom { visibility: hidden; } .MJXp-munderover { display: inline-table !important; } .MJXp-over { text-align: center; display: inline-block !important; } .MJXp-over > * { display: block !important; } .MJXp-munderover > * { display: table-row !important; } .MJXp-mtable { vertical-align: 0.25em; margin: 0px 0.125em; } .MJXp-mtable > * { vertical-align: middle; display: inline-table !important= ; } .MJXp-mtr { display: table-row !important; } .MJXp-mtd { text-align: center; padding: 0.5em 0px 0px 0.5em; display: tabl= e-cell !important; } .MJXp-mtr > .MJXp-mtd:first-child { padding-left: 0px; } .MJXp-mtr:first-child > .MJXp-mtd { padding-top: 0px; } .MJXp-mlabeledtr { display: table-row !important; } .MJXp-mlabeledtr > .MJXp-mtd:first-child { padding-left: 0px; } .MJXp-mlabeledtr:first-child > .MJXp-mtd { padding-top: 0px; } .MJXp-merror { background-color: rgb(255, 255, 136); color: rgb(204, 0, 0);= border: 1px solid rgb(204, 0, 0); padding: 1px 3px; font-style: normal; fo= nt-size: 90%; } .MJXp-scale0 { transform: scaleX(0); } .MJXp-scale1 { transform: scaleX(0.1); } .MJXp-scale2 { transform: scaleX(0.2); } .MJXp-scale3 { transform: scaleX(0.3); } .MJXp-scale4 { transform: scaleX(0.4); } .MJXp-scale5 { transform: scaleX(0.5); } .MJXp-scale6 { transform: scaleX(0.6); } .MJXp-scale7 { transform: scaleX(0.7); } .MJXp-scale8 { transform: scaleX(0.8); } .MJXp-scale9 { transform: scaleX(0.9); } .MathJax_PHTML .noError { font-size: 90%; text-align: left; color: black; p= adding: 1px 3px; border: 1px solid; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC---- Content-Type: text/css Content-Transfer-Encoding: quoted-printable Content-Location: cid:css-36633f98-5e25-4bbb-996b-440de60e4272@mhtml.blink @charset "utf-8"; h1, h2, h3 { text-align: center; } ------MultipartBoundary--Tq8vgAGBKLw0SU6jIxiajdX0vv8evHDemoB6LydDhC------